lemmaroutine

Local comparison of elliptic index with integer

Elliptic tests force

v_q(\widetilde a-\eta(a))\ge K

for all

a\in R

turns elliptic conditions into a congruence mod

q^K

Lemma·red:local-comparison·§2.1 Elliptic indices and the local comparison

Statement (paper)

Let

K\ge1

be an ordinary integer. Let

q

be a prime of the ambient integer sort

{}^*\Z

, possibly nonordinary, with

q\notin S_E

and

q>K+1

, and let

v_*

be a place of

{}^*F

above

q

, normalized by

v_*(q)=1

. Suppose there are nonzero

h,j_0\in R

, chosen independently of

a

, such that for every

a\in R

: \begin{enumerate}[label=(\roman*),leftmargin=*] \item The points

\widetilde T_h,\widetilde T_{ha}

reduce to

O

v_*

, the value

\ell_K(z(\widetilde T_h))

is nonzero, and

v_*\!\left( \frac{\ell_K(z(\widetilde T_{ha}))}{\ell_K(z(\widetilde T_h))} -\widetilde a\right)\ge K.

\item The points

\widetilde T_{j_0}^{\sigma}

and

\widetilde T_{(h-4)a}^{\sigma}

reduce to

O

on the conjugate curve at the same place

v_*

, and

v_*\!\left( \frac{z(\widetilde T_{(h-4)a}^{\sigma})} {z(\widetilde T_{j_0}^{\sigma})}\right)\ge K.

\end{enumerate} Then every

\widetilde a

is integral at

q

, and

v_q(\widetilde a-\eta(a))\ge K \qquad(a\in R).

Here

z(O)=0

. The denominator in (ii) is nonzero because

j_0\ne0

and the nonzero points in have nonzero

x

- and

y

-coordinates.

In plain English

Under the stated hypotheses at a place

v_*

above a good prime

q>K+1

, if the ratio

\ell_K(z(\widetilde T_{ha}))/\ell_K(z(\widetilde T_h))

approximates

\widetilde a

to precision

K

and the conjugate-curve ratio

z(\widetilde T^{\sigma}_{(h-4)a})/z(\widetilde T^{\sigma}_{j_0})

has valuation at least

K

, then each

\widetilde a

is integral at

q

and

v_q(\widetilde a-\eta(a))\ge K

for all

a\in R

Why it's here

This is the local step that converts the elliptic-curve conditions (i) and (ii) into a congruence modulo

q^K

between

\widetilde a

and the index

\eta(a)

. It uses the formal group comparison on both

E

and the conjugate curve

E^\sigma

to read off the

q

-adic information, and it feeds the later results on integer coverage and index congruences.

Classified routine:

Direct local consequence of the formal group comparison (ell-formal), which reads q-adic valuations of elliptic multiples off the formal logarithm; the argument is bookkeeping of valuations on E and its conjugate curve E^σ.

Uses

propFormal group comparison of elliptic multiples

Used by

corInteger coverage by prime patternspropIndex congruence at every contact prime

Figures

A proof-flow diagram of six boxes connected by arrows. The left column starts with varying $A\in R$ and choosing a representation at precision $K$ with slopes $\widetilde s_i$, then passes to local analysis at each prime $q\mid M_K(\widetilde s_i)$ (pole parity and coarsening), then to the paired elliptic tests giving $v_q(\widetilde a-\eta(a))\ge K$. The right column keeps the root tuple fixed with nonzero $\delta=f(\eta(\mathbf a))$ bounded by $C_fB^D$; an arrow labeled 'polynomial evaluation' and an arrow from the tests feed a box concluding $M_K(\widetilde s_i)^K\mid\delta$. A final wide box applies the five-point height estimate to get $h(\widetilde A)\ll_f B$ uniformly for $A\in R$. — Figure 1diagramp. 16
The use of one nonzero $\delta$ for every $A\in R$ . All slopes shown come from the representation chosen at the fixed ordinary precision $K$ , and $M_K$ uses the fixed exceptional set for $K$ . The comparison holds for every $a\in R$ , so it applies to the fixed root tuple. The implied constant in the final bound is ordinary and independent of $A$ and $\mathbf a$ .
What it shows
A proof-flow diagram of six boxes connected by arrows. The left column starts with varying $A\in R$ and choosing a representation at precision $K$ with slopes $\widetilde s_i$ , then passes to local analysis at each prime $q\mid M_K(\widetilde s_i)$ (pole parity and coarsening), then to the paired elliptic tests giving $v_q(\widetilde a-\eta(a))\ge K$ . The right column keeps the root tuple fixed with nonzero $\delta=f(\eta(\mathbf a))$ bounded by $C_fB^D$ ; an arrow labeled 'polynomial evaluation' and an arrow from the tests feed a box concluding $M_K(\widetilde s_i)^K\mid\delta$ . A final wide box applies the five-point height estimate to get $h(\widetilde A)\ll_f B$ uniformly for $A\in R$ .
Read from the image by claude-opus-5.5 · confidence 85%
SupportspropFinite test success gives integer zeroclaimPolynomial error delta is zero or boundedclaimContact radical K-th power divides deltaclaimUniform height bound for ring elementspropIndex congruence at every contact primelemLocal comparison of elliptic index with integerlemOdd valuation selects a tested branchthmHeight bounded by radical of contact primes

Sources

PaperOpen

Section: §2.1 Elliptic indices and the local comparison
Page: —
Label: red:local-comparison
Line: sections/02-reduction.tex:143

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